Questions tagged [copula]

A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the unit interval.

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Finding the expected value of two correlated RVs

$\newcommand{\Exp}[1]{\mathbb{E}\left[#1\right]}$ I am interested in understanding wether the following approach holds when calculating the expectation of two correlated random variables. Suppose $X\...
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Kendall 's tau in high dimensional setting

Often I see that the Kendall's tau is defined for 2 random variables by using their bivariate copula $C(u_1, u_2)$ I wonder in the case of multidimensional setting (i.e. from 3 dimensions), how the ...
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Proving that copulas are Lipschitz continuous

A copula is a function $C:[0,1]^2\to[0,1]$ such that $C(x,0)=C(0,x)=0$ for all $x\in[0,1]$, $C(x,1)=C(1,x)=x$ for all $x\in[0,1]$, and \begin{equation}\label{ineq} C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(...
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Calculating the joint cumulative distribution function from a junction tree

Assume I have the following Junction Tree between random variables $X_1,\dots,X_7$ that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ...
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Product of correlated random variables and its transformation

There is an interesting result, saying that if $Z_1, Z_2$ are standard normal random variables with a correlation $\rho\in (-1,1)$, then the product $Z=Z_1Z_2$ has a density function explicitly given ...
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What is the origin of the name "elliptical" copula?

Could you please explain me why the eliptical copula is called "eliptical" ? I have read many articles but the authors assumed the knowledge of where the elliptical copula comes from is ...
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Combining conditional probabilities using an unconditional copula

First: just a bit of background on copulas. Suppose we have a pair of continuous random variables $Y_1, Y_2$ with distribution functions $F_1(y_1)=P(Y_1\leq y_1)$ and $F_2(y_2)=P(Y_2\leq y_2)$. Let ...
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Kendall's Tau of FGM Copula

The Fairly-Gumbel-Morgenstern Copula is defined as $C_\theta(u,v) = uv + \theta uv(1-u)(1-v)$ With (u,v) $\in$ $[0,1]^2$ and $\theta \in [-1,1]$ in order to $C_\theta$ to be a copula. I know that from ...
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Proving that $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$ is a copula

I have the following problem. I need to prove the copula property for the function $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$, better known as the lower Fréchet-Hoeffding-Boundary. A copula is defined as a ...
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Frequently used methods to estimate copula of discrete random variable in high dimension

I am new to the estimation of copula with discrete marginal distributions. For example, I have the data of 30 discrete random count variables from $X_1$ to $X_{30}$, some $X_j$ has Poisson ...
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Question on the Farlie–Gumbel–Morgenstern copula

To compute the joint bivariate probability density function (PDF) given marginal PDFs we can use the copula. The Sklar's theorem imply that copulas can be used to express a multivariate distribution ...
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Uniqueness of copula for discrete random variable explanation

The Sklar's theorem of copula states that Let $H$ be a two-dimensional distribution function with marginal distribution functions $F$ and $G$. Then there exists a copula $C$ such that $H(x,y)=C(F(x),...
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Flipping positive quadrant dependent copula into negative quadrant dependent

Most copula families (e.g. Archimedean copulae) seem to only parameterize positive quadrant dependent (PQD) copulae. Suppose we have a PQD copula $C(u, v)$. Can we obtain a negative quadrant dependent ...
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Marshall -olkin Copula and absolutely continuous

the Marshall -olkin copula is given by $$C(x,y)= \min(x^{1/3}y , xy^{3/7}).$$ I wonder if this copula has a density function, because I had tried find density of MO copula in R: ...
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Kendall’s tau between $X$ and $1/Y$

I'm currently studying for my statistics exam by doing exercises from the book Statistics and Data Analysis for Financial Engineering with R examples. I'm struggling with this exercise from chapter 8: ...
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Proof that Pearson correlation is/isn't a functional of the copula for a given pair of random variables

I have a growing interest and respect for the subject of copulas initially thanks to comments made on stats.SE by kjetil-b-halvorsen. The most interesting to me right now is the following: "[T]...
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Sampling from Gaussian Copula with a conditional distribution approach

In the paper "Cheng and al. (2007)" on pages 193-194, the authors propose an algorithm that generates variables with a given copula function $C$ being the joint distribution. This procedure ...
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Life expantancy of correlated light bulb batches

I have 3 light bulbs from different build batches, they have a rate of failure of 0.02, 0.05 and 0.09 each month. The failure rate between the batches is correlated, the failures are normally ...
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Showing that the lower bivariate Fréchet-Hoeffding bound is a copula.

I want to show that the bivariate Fréchet-Hoefdding lower bound is indeed a copula. The bivariate function is defined by: $W(x,y)=\max\{x+y-1,0\}, \ x,y\in[0,1] $ Definition "Copula": A two-...
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How to deal with copulas?

I want to model a couple of variables $(X,Y)$ using copulas. My idea is to model the marginal distributions and then use copulas to combine marginal distributions and copula to get a joint ...
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Kendall's tau for archimedian copula

Consider a 2-dimensional archimedian copula with generator $\psi$ and $\gamma:=\psi^{-1}$, i.e. $$C(u,v)=\gamma(\psi(u)+\psi(v))$$ I want to show that $$\tau=1-\int_{0}^{\infty}t\gamma'(t)^2dt$$ I ...
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Find $\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x)$ where $X,Y$ poisson

Let $X=Y_1+N,Y=Y_2+N$ where $N\sim \text{pois}(\lambda), Y_1\sim \text{pois}(\lambda_1), Y_2\sim \text{pois}(\lambda_2)$ and $N,Y_1,Y_2$ are independent. I'm asked to find $$\lim_{x\uparrow 1}\mathbb{...
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$P(u_1\leq\alpha,u_2\leq\beta)$ and $P(u_1>\alpha,u_2>\beta)$ and copulas

I know this is meant to be elementary, but I don't know why I cannot convince myself -- given a copula $C(u_1,u_2)=P(u_1\leq \alpha,u_2\leq\beta)$, why is $P(u_1>\alpha,u_2>\beta)=1-\alpha-\beta-...
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Verifying if a function is 2-increasing

In the book An Introduction to Copulas (2006), by Roger B. Nelsen, on page 8, the author defines a function $H$ as 2-increasing if \begin{equation} V_H(B)=H(x_2,y_2)-H(x_2,y_1)- H(x_1,y_2)+H(x_1,y_1)\...
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How can I get an intuition about Copula?

I am really struggling with getting an intuition about copulas. I have red many articles and I am stuck at what is the concept/idea behind it. For example if I have two random variables X and Y and I ...
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Correlation matrix of Gaussian copula

The question is related to the Gaussian copula. Let $\Phi(x)$ denote the cdf of standard Normal distribution. Let $(X_1, X_2) \sim \mathcal{N}(0,\Sigma)$ be joint Normal with covariance matrix $$ \...
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Joint Distribution of Two Dependent Variables having the Marginals

From two Independent Normal Random Variables: $X \sim N(\mu_1,\sigma) $ and $Y\sim N(\mu_1,\sigma)$, I created two DEPENDENT random variables $Z$ and $W$: $Z= X - Y$ $W= X - g(Y)$ where $g(\...
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How to compare the entropy of copulas?

Three different bivariate copulas are shown below with increasing degrees of dependence (parameter $\theta$). Differential entropy is a measure of disorder in a probability density like the copula. ...
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Impact of copula choice on quantiles (sum of random variables)

I am trying to get my head around the impact of different dependence structures (copulas) on the risk (quantiles) of a sum of dependent random variables (with arbitrary marginals). In a multivariate ...
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Why is copula entropy negative?

Most statistical measures are non-negative. Shannon entropy, $H(X)$, a statistical measure of disorder or uncertainty in probability distributions, is also non-negative, despite it having a negative ...
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Why do elliptical copula densities contain $x_1$ and $x_2$, but Archimedean copula densities contain $u_1$ and $u_2$?

$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$ is the ...
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How to simplify out of the gamma function $\Gamma(z)$?

$$\mathrm{d}t(x_i; \nu) = \frac{\Gamma \bigg(\frac{\nu+1}{2}\bigg) }{\Gamma (\frac{\nu}{2}) \sqrt{\pi\nu}} \Bigg( 1+\frac{x_i^2}{\nu} \Bigg)^{-\frac{\nu+1}{2} } \enspace, i=1,2$$ How to derive a ...
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Derivation of bivariate Gaussian copula density

The multivariate Gaussian copula density, derived here, is $$c(u_1,\ldots,u_n;\Sigma)=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right)$$ where $\Sigma$ is the covariance ...
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Integrate $\int_{0}^{1} u^{\frac{\alpha}{\beta}-\alpha}(1-\alpha) \log \left((1-\alpha) u^{-\alpha}\right) d u $

Where $c(u, v)=u^{-\alpha}(1-\alpha) $ is the Marshall-Olkin copula density, we have the following integral: \begin{align} I_{1} &=\iint_{E} c(u, v) \log c(u, v) d u d v \\ &=\int_{0}^{1} \...
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Entropy of the bivariate Gaussian copula: Closed-form analytical solution

Background The bivariate Gaussian copula function is $$C_{\rho}(u,v)=∫_{-∞}^{Φ^{-1}(u)}∫_{-∞}^{Φ^{-1}(v)}\frac{1}{2π\sqrt{1-ρ^2}}×exp⁡(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})dxdy.$$ The multivariate Gaussian ...
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How to simplify function to $\log (1-\theta)+\frac{\theta}{2-\theta}$?

\begin{align} f(\theta)& = \log \frac{1}{1-\theta}-\frac{\theta}{2-\theta}+\frac{\theta^{2}}{(2-\theta)^{2}} \\ & = \log (1-\theta)+\frac{\theta}{2-\theta} \end{align} How are the two ...
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Do copulas preserve uniform convergence?

Suppose I have a distribution $Q$ of $\left(Y_i, X_i\right)$, such that $Q(y,x)=C(F_Y(y), F_X(x))$, for some copula $C$ and marginal CDFs $F_X$ and $F_Y$, by Sklar's theorem. Now, suppose I have ...
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Showing that a function is a (family of) copula(s)

I am currently working with the book An Introduction to Copulas by Roger Nelsen. I am having trouble solving one of the exercises in the first chapter. The exercise reads as follows: Let $C$ be a ...
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Simplify $\mathbb{E}\left[ -\log c(u,v)\right] $, the expected logarithm of the copula density

2020 11.26: I finished the derivation but haven't posted it here. I leave this open so that those willing to try their own version can if they want Question How can we derive a closed-form analytical ...
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Archimedean Clayton copula entropy

Question I would like to derive the entropy of Archimedean parametric copulas (Clayton, Frank, or Gumbel), focusing here on the Clayton copula. Link to similar question on the t-copula. The bivariate ...
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Gluing copulas along a curve

Let $C(u,v)$ be a 2-copula and $W(u,v)=\max(u+v-1,0)$. I want to glue $C$ with $W$ along the curve $u^2+v^2=1$ s.t $$D(u,v)=\begin{cases}u+v-1,& u^2+v^2\ge 1,\\ C(u,v), &u^2+v^2<1 \end{...
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Find a joint distribution function with known Kendall's tau (Copula)

Assume we have two random variables $X_1\sim Exp(2)$, $X_2\sim Exp(1/2)$. I need to find a joint distribution function, so that Kendall's Tau will be $\rho_\tau(X_1,X_2)=-0.85$. I know that somehow I ...
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What is the correlation of a copula of a multivariate normal distribution? [duplicate]

If $X_1, X_2 \sim \mathcal{N}(0,1)$ with correlation $\rho$, and $Y_1 = F_{X_1}(X_1)$ and $Y_2 = F_{X_2}(X_2)$, what is the correlation between $Y_1$ and $Y_2$? What is a proof that the correlation of ...
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Copula with a certain correlation

What does it mean for values to be "drawn from a normal copula with correlation $\rho\in [0, 1]$"? Is that a normal distribution with a covariance matrix whose entries are uniform random in $...
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Archimedean Copula Probabilities

I have the following excercise : Let C be an Archimedean copula with generator given by $\psi(x) = E[e^{ −xV} ]$, where V is an exponentially distributed random variable with expectation 1. Calculate ...
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Copula Theory : CDF from Marginals

I have given $(X,Y)$ to be a two-dimensional random vector with Exp(1)-marginals and a Copula $C(u,v) = uv + (1-u)(1-v)uv$ I need to determine the density of $(X,Y)$, if it exists. I would assume that ...
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Calculating Kendall's Tau (Copulas)

Let $(X_1,X_2)$ be a random vector with $X_1 \sim Exp(1)$ and $X_2 \sim N(0,1)$. The dependence structure is given by the copula $$C(u_1,u_2)=\frac{1}{3}W(u_1,u_2)+\frac{2}{3}\Pi(u_1,u_2), \ \ \ u \in ...
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Joint probability from marginal and relation between variables

I would like to know whether it is possible to obtain the joint density function $p(x,y)$ if I know one marginal, which is Gaussian $p(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp(-\frac{1}{2} (\frac{x - \mu}...
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Gaussian and Student-t copulas derivatives

I am working on a data set with the Gaussian and the Student-t copulas and I need to define their derivatives. For the Gaussian copula I have defined it as follows: $$ \frac{d}{du}C^{Gauss}_{\alpha}= \...
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Joint distribution of two uniform random variables

Consider two uniform random variables $U, V$ . If $ U \sim \mbox{Uniform}(0,1)$ and $V = 1 -U$ how would we show that the joint distribution of $(U, V)$ is given by $\max \{ u + v -1,0 \}$? The ...
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